Symmetric Kneser's Theorem with Trios and 3-Transform

摘要

We give a new equivalent restatement and a new proof in terms of trios to the classical Kneser's theorem. In the finite case, our restatement takes the following, particularly symmetric shape: if A, B, and C are subsets of a finite abelian group G such that A + B + C ≠ G, then, denoting by H the period of the sumset A + B + C, we have |A| + |B| + |C|≤ |G| + |H|. The proof is based on an extension of the familiar Dyson transform onto set systems containing three (or more) sets.